Channel estimation for wireless systems without matrix inversion

ABSTRACT

In various embodiments, techniques are provided to determine channel characteristics of various communication systems such as OFDM systems or systems using a plurality of transmit antennas by using various sets of training symbols that produce zero cross-correlation energy. Channel communication can accordingly be simplified as the zero cross-correlation property allows for channel estimation without a matrix inversion.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.14/258,849, filed on Apr. 22, 2014, which is a continuation of U.S.patent application Ser. No. 13/925,306, filed on Jun. 24, 2013, whichhas issued as U.S. Pat. No. 8,724,725, which is a continuation of U.S.patent application Ser. No. 12/550,122, filed on Aug. 28, 2009, whichhas issued as U.S. Pat. No. 8,472,553, which is a continuation of U.S.patent application Ser. No. 11/931,922 filed on Oct. 31, 2007, which hasissued as U.S. Pat. No. 7,583,761, which is a continuation of U.S.patent application Ser. No. 11/508,480 filed on Aug. 22, 2006, which hasissued as U.S. Pat. No. 7,305,051, which is a continuation of U.S.patent application Ser. No. 09/862,755 filed on May 21, 2001, which hasissued as U.S. Pat. No. 7,103,115. All of the above cited applicationsare herein incorporated by reference in their entirety.

BACKGROUND OF THE INVENTION

Field of Invention

This invention relates to channel estimation in wireless systems.

Description of Related Art

As wireless communications systems are deployed around the world, theimportance of providing clear and cost-effective communication servicesincreases. Unfortunately, providing clear communications can requiremitigating various obstacles such as inter-symbol-interference (ISI). Toreduce ISI, a technique known as orthogonal frequency divisionmultiplexing (OFDM) can be used. Orthogonal frequency divisionmultiplexing is a communication paradigm where a single communicationchannel is divided into many narrow sub-bands, which then can betransmitted in parallel. By transmitting symbols in this fashion, theduration of each symbol can be dramatically increased, which can greatlyreduce or completely eliminate ISI problems.

Unfortunately, individual sub-bands within an OFDM transmission aresubject to Rayleigh fading, especially when used in mobile communicationsystems. While the effects of Rayleigh fading can be mitigated by usingmultiple transmitter and/or receiver antennas, estimating the channelcharacteristics for all transmitter-receiver antenna pairs can bedifficult and computationally intensive. Accordingly, there is a needfor apparatus and techniques that provide for better channel estimation.

SUMMARY OF THE INVENTION

In various embodiments, techniques are provided to determine channelcharacteristics of various communication systems, such as OFDM systemsor systems using a plurality of transmit antennas.

In a first embodiment, methods and apparatus for transmitting trainingsymbols based on a set of first training symbols and one or more sets ofsecond training symbols using one or more second communication channelsis provided where a cross-correlation between the first set of trainingsymbols and at least one of the sets of second training symbols isessentially zero.

In a second embodiment, methods and apparatus for channel estimation areprovided by first receiving a first set of training symbols related to afirst antenna and a set of second training symbols related to a secondantenna and then estimating various communication channels. When thecross-correlation between the sets of training symbols is essentiallyzero, channel estimation can be achieved without a matrix inversion,thus simplifying channel estimation.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is described in detail with regard to the followingfigures, wherein like numerals reference like elements, and wherein:

FIG. 1 is a block diagram of an exemplary communication system;

FIG. 2 depicts an OFDM signal having multiple sub-bands;

FIG. 3 depicts an exemplary communication signal of an OFDM sub-band;

FIG. 4 is a block diagram of an exemplary OFDM encoder;

FIG. 5 is a block diagram of an exemplary training symbol generator;

FIG. 6 is a block diagram of an equalizer with an exemplary channelestimator; and

FIG. 7 is a flowchart outlining an exemplary technique for generatingand communicating with sets of training symbols.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

This non-provisional application incorporates the substance of U.S.patent application Ser. No. 09/861,811 entitled “CHANNEL ESTIMATION FORWIRELESS SYSTEMS” to Ye LI (Attorney Docket No. 107640, 1999-0759A) nowU.S. Pat. No. 7,012,966. The above non-provisional application isconcurrently filed and commonly assigned and hereby incorporated byreference in its entirety including all references cited therein.

For wireless systems, channel estimation can be difficult andcomputationally intensive. Generally, channel estimation for wirelesssystems can be performed by embedding a pattern of known symbols calledtraining symbols within a transmitted signal and comparing the embeddedtraining symbols against an expected pattern of training symbols at areceiver.

However, for communication systems such as orthogonal frequency divisionmultiplexed (OFDM) systems having multiple sub-bands, each sub-band mustbe estimated, thus increasing computational requirements. When an OFDMcommunication system uses multiple transmit antennas and/or multiplereceive antennas, each sub-band for each communication channel must beestimated, thus further increasing computational requirements.Fortunately, the problem of estimating the channel parameters fordifferent sub-bands in an OFDM channel can be simplified based on thecorrelated nature of the sub-bands. Examples of such systems can befound in Yi, L., Seshardi, N. and Ariyavisitakul, S., “Channelestimation for OFDM systems with transmitter diversity in mobilewireless channels”, IEEE Journal of Selected Areas in Comm., Vol. 17,pp. 461-471 (March 1999) incorporated herein by reference in itsentirety. Unfortunately, regardless of any advantages posed byintra-channel simplification for a single OFDM channel, the problem ofestimating different OFDM channels remains.

However, in systems using multiple transmit antennas, by carefullychoosing different patterns of training symbols such that each patternof training symbols for any transmit antenna has zero cross-correlatedenergy with the training symbol pattern of any other transmit antenna,channel estimation can be greatly simplified. Such judicious choices oftraining symbols not only can reduce computational complexity, but alsocan provide the most accurate channel estimates possible.

FIG. 1 is a block diagram of an exemplary transmission system 100. Thetransmission system 100 includes an encoder 110 having a number ofassociated OFDM transmitters 120-1, 120-2, . . . 120-N and respectivetransmit antennas 130-1, 130-2, . . . 130-N, and an equalizer 160 havinga number of associated OFDM receivers 150-1, 150-2, . . . 150-M withrespective receive antennas 140-1, 140-2, . . . 140-M.

In operation, the encoder 110 can form blocks of symbols that can beprovided to the various OFDM transmitters 120-1, 120-2, . . . 120-N. Thevarious OFDM transmitters 120-1, 120-2, . . . 120-N, in turn, canmodulate the blocks of symbols into electromagnetic carriers such asradio-frequency waves that can then be transmitted using theirrespective transmitting antennas 130-1, 130-2, . . . 130-N. The variousradio-frequency signals 135 can then be received by the receive antennas140-1, 140-2, . . . 140-M and fed to their respective OFDM receivers150-1, 150-2, . . . 150-M. The OFDM receivers 150-1, 150-2, . . . 150-Mcan then transform the received radio-frequency signals 135 intobase-band signals, digitize the base-band signals and provide thedigitized base-band signals to the equalizer 160. The equalizer 160, inturn, can extract the symbols from the digitized base-band signals andperform various operations on the symbols.

As shown in FIG. 1, the radio-frequency signals 135 transmitted by eachtransmit antenna 130-1, 130-2, . . . 130-N can be subsequently receivedby each of the receiving antennas 140-1, 140-2, . . . 140-M. While FIG.1 depicts the various communication channels as single direct pathsbetween each transmit/receive antenna pair, it should be appreciatedthat each radio-frequency signal 135 can propagate from each transmitantenna 130-1, 130-2, . . . 130-N to each receive antenna 140-1, 140-2,. . . 140-M not only through a direct path, but can also propagate fromeach transmit antenna 130-1, 130-2, . . . 130-N to each receive antenna140-1, 140-2, . . . 140-M through a variety of indirect paths (notshown).

The various radio-frequency signal paths for a particulartransmit/receive antenna pair can produce a complex communicationchannel, which can be distinctly different from any other communicationchannel defined by another transmit/receive antenna pair. Generally, thechannel characteristics of an individual mobile wireless channel i.e.,the impulse response, can be described by Eq. (1):

$\begin{matrix}{{{h\left( {t,\tau} \right)} = {\sum\limits_{k}{{\gamma_{k}(t)}{c\left( {\tau - \tau_{k}} \right)}}}},} & (1)\end{matrix}$

where τ_(k) is the delay of the k-th path, γ_(k)(t) is the correspondingcomplex amplitude for the k-th path, and c(t) is the shaping pulse forthe k-th path whose frequency response is usually a square-root raisedcosine Nyquist filter. When a communication channel is a mobile wirelesschannel, the motion of a vehicle can affect the complex amplitudesγ_(k)(t)'s making each complex amplitude γ_(k)(t) a wide-sensestationary (WSS), narrow-band complex Gaussian process that can beindependent for different signal paths.

From Eq. (1), the frequency response H(t, f) of a communication channelat time t can be described by Eq. (2):

$\begin{matrix}{{H\left( {t,f} \right)}\overset{\Delta}{=}{\int_{- \infty}^{+ \infty}{{h\left( {t,\tau} \right)}{\exp \left( {{- {j2\pi}}\; f\; \tau} \right)}\ {\tau}\mspace{14mu} {or}}}} & (2) \\{{\overset{\Delta}{=}{{C(f)}{\sum\limits_{k}{{\gamma_{k}(t)}{\exp \left( {{- {j2\pi}}\; f\; \tau_{k}} \right)}}}}}{where}} & (3) \\{{{C(f)}\overset{\Delta}{=}{\int_{- \infty}^{+ \infty}{{c(\tau)}{{\exp \left( {{- {j2\pi}}\; f\; \tau_{k}} \right)}.}}}}\ } & (4)\end{matrix}$

For OFDM systems with proper cyclic extension and timing, the channelfrequency response can be expressed by Eq.(5):

$\begin{matrix}{{{H\left\lbrack {n,k} \right\rbrack}\overset{\Delta}{=}{{H\left( {{nT}_{f},{k\; \Delta \; f}} \right)} = {\sum\limits_{L = 0}^{K_{o} - 1}{{h\left\lbrack {n,l} \right\rbrack}W_{K}^{kl}}}}},} & (5)\end{matrix}$

where h[n, l]

h(nT_(f), kT_(f)/K), W_(K)=exp(−j2π/K), K is the number of sub-bands(tones) in an OFDM block, T_(f) is the block length and Δf is thesub-band (tone) spacing.

Following Eq. (5), the frequency response at the k-th tone of an n-thblock of OFDM symbols corresponding to an i-th transmit antenna can beexpressed by Eq.(6):

$\begin{matrix}{{H_{i}\left\lbrack {n,k} \right\rbrack} = {\sum\limits_{l = 0}^{K_{o} - 1}{{h_{i}\left\lbrack {n,l} \right\rbrack}{W_{K}^{kl}.}}}} & (6)\end{matrix}$

Equation (6) demonstrates that Hi[n, k] can be obtained by estimating orotherwise acquiring h_(i)[n, k]. Accordingly, the received signal r[n,k] at each receive antenna 140-1, 140-2, . . . 140-M can be expressed byEq. (7):

$\begin{matrix}{{{r\left\lbrack {n,k} \right\rbrack} = {{\sum\limits_{i = 1}^{M}{{H_{i}\left\lbrack {n,k} \right\rbrack}{t_{i}\left\lbrack {n,k} \right\rbrack}}} + {w\left\lbrack {n,k} \right\rbrack}}},} & (7)\end{matrix}$

where M is the number of transmit antennas, k denotes a particular OFDMsub-band and k=0, 1, . . . , K−1 for all n blocks. If the transmittedsignals t_(i)[n, k]'s from each transmit antenna contain known signalssuch as training symbols, the temporal estimation of the variouscommunication channels h_(i)[n,l]'s can be derived using Eq. (8):

$\begin{matrix}{{{\begin{pmatrix}{Q_{11}\lbrack n\rbrack} & {Q_{12}\lbrack n\rbrack} & \ldots & {Q_{1P}\lbrack n\rbrack} \\{Q_{21}\lbrack n\rbrack} & {Q_{22}\lbrack n\rbrack} & \; & \vdots \\\vdots & \; & \ddots & \; \\{Q_{P\; 1}\lbrack n\rbrack} & \ldots & \; & {Q_{PP}\lbrack n\rbrack}\end{pmatrix}\begin{pmatrix}{{\overset{\sim}{h}}_{1}\lbrack n\rbrack} \\{{\overset{\sim}{h}}_{2}\lbrack n\rbrack} \\\vdots \\{{\overset{\sim}{h}}_{P}\lbrack n\rbrack}\end{pmatrix}} = \begin{pmatrix}{p_{1}\lbrack n\rbrack} \\{p_{2}\lbrack n\rbrack} \\\vdots \\{p_{P}\lbrack n\rbrack}\end{pmatrix}},{or}} & (8) \\{{\begin{pmatrix}{{\overset{\sim}{h}}_{1}\lbrack n\rbrack} \\{{\overset{\sim}{h}}_{2}\lbrack n\rbrack} \\\vdots \\{{\overset{\sim}{h}}_{P}\lbrack n\rbrack}\end{pmatrix} = {\begin{pmatrix}{Q_{11}\lbrack n\rbrack} & {Q_{12}\lbrack n\rbrack} & \ldots & {Q_{1P}\lbrack n\rbrack} \\{Q_{21}\lbrack n\rbrack} & {Q_{22}\lbrack n\rbrack} & \; & \vdots \\\vdots & \; & \ddots & \; \\{Q_{P\; 1}\lbrack n\rbrack} & \ldots & \; & {Q_{PP}\lbrack n\rbrack}\end{pmatrix}^{- 1}\begin{pmatrix}{p_{1}\lbrack n\rbrack} \\{p_{2}\lbrack n\rbrack} \\\vdots \\{p_{P}\lbrack n\rbrack}\end{pmatrix}}},{where}} & (9) \\{{{{\overset{\sim}{h}}_{i}\lbrack n\rbrack}\overset{\Delta}{=}\left( {{{{\overset{\sim}{h}}_{i}\left\lbrack {n,0} \right\rbrack}\mspace{14mu} \ldots}\mspace{14mu},{{\overset{\sim}{h}}_{i}\left\lbrack {n,{K_{o} - 1}} \right\rbrack}} \right)^{T}},} & (10) \\{{{Q_{ij}\lbrack n\rbrack}\overset{\Delta}{=}\left( {q_{ij}\left\lbrack {n,{i - j}} \right\rbrack} \right)_{i,{j = 1}}^{K_{o}}},{and}} & (11) \\{{{p_{i}\lbrack n\rbrack}\overset{\Delta}{=}\left( {{p_{i}\left\lbrack {n,0} \right\rbrack},\ldots \mspace{14mu},{p_{i}\left\lbrack {n,{K_{o} - 1}} \right\rbrack}} \right)^{T}},} & (12)\end{matrix}$

where {tilde over (h)}_(i)[n] is the estimated channel impulse responsefor the channel between an i-th transmit antenna and a particularreceive antenna, Q_(ij)[n] is a measure of the correlated energy betweenthe set of training symbols t_(i)[n, k] of an i-th transmit antenna andthe set of training symbols t_(j)[n, k] of a j-th transmit antenna andp_(i)[n] is the cross-correlation vector between the set of trainingsymbols t_(i)[n, k] of an i-th transmit antenna and a received signalr[n, k] at a particular receive antenna, and where:

$\begin{matrix}{{{q_{ij}\left\lbrack {n,l} \right\rbrack}\overset{\Delta}{=}{\sum\limits_{k = 0}^{K - 1}{{t_{i}\left\lbrack {n,k} \right\rbrack}{t_{j}^{*}\left\lbrack {n,k} \right\rbrack}W_{K}^{- {kl}}}}},{and}} & (13) \\{{p_{i}\left\lbrack {n,l} \right\rbrack}\overset{\Delta}{=}{\sum\limits_{k = 0}^{K - 1}{{r\left\lbrack {n,k} \right\rbrack}{t_{i}^{*}\left\lbrack {n,k} \right\rbrack}{W_{K}^{- {kl}}.}}}} & (14)\end{matrix}$

Equation (13) provide a measure of the cross-correlation of energybetween two sets of training symbols, i.e., a cross-correlationestimate, and Eq. (14) provides a similar measure between a particularset of training symbols and a received signal. While the particularcross-correlation estimates can be derived using Eqs. (13) and (14), itshould be appreciated that any substantially accurate measure ofcorrelation energy derived by any known or later developed means can beused without departing from the spirit and scope of the presentinvention.

From Eqs. (8) and (9), one can see that a matrix inversion is requiredto get the temporal estimation of each channel impulse response h_(i)[n,k], i.e., {tilde over (H)}=Q⁻¹P.

FIG. 2 depicts an exemplary OFDM signal 200 displayed along a time axis210 and against a frequency axis 220. As shown in FIG. 2, the OFDMsignal 200 contains a number of individual sub-bands (tones) 230-1,230-2, . . . 230-K with each respective sub-band centered on arespective one of closely spaced frequencies f₁, f₂, f_(K). FIG. 3depicts an exemplary communication signal 300 capable of being embeddedin the various sub-bands of FIG. 2. As shown in FIG. 3, thecommunication signal 300 contains a number of sync symbols 310, a numberof training symbols 320, a number of data symbols 330 and a number ofguard symbols 340.

Data symbols, also known as payload symbols, can contain information tobe transmitted. Guard symbols are symbols that can pad either or both ofthe beginning and end of a burst transmission and can be used for avariety of purposes including providing buffering, timing andsynchronization. Sync symbols are predetermined symbols placed atvarious strategic positions within a block of data that can allow areceiver to synchronize or otherwise extract timing information from atransmitted signal.

Training symbols, like sync symbols, can be predetermined symbols placedat known positions. However, unlike sync symbols, training symbols areusually configured to enable an equalizer to estimate a givencommunication channel. It should be appreciated that, in variousexemplary embodiments, the training symbols 320 can be any set ofsymbols suitable for training an equalizer. For example, the exemplarytraining symbols 320 can be formed taking into account various factorssuch as their suitability for clock recovery, frequency-shiftestimation, their peak-to-average ratio of signal strength or any otherknown or later recognized factor useful for generating an advantageousor otherwise adequate training sequence.

While the exemplary communication signal 300 is a burst signal with aparticular form, it should be appreciated that the form of a burstsignal can vary without departing from the spirit and scope of thepresent invention. For example, it should be appreciated that thetraining symbols 320 can be dispersed intermittently within the payloadsymbols 330. It should further be appreciated that while the exemplarycommunication signal 300 is a burst signal, the communication signal 300can take various other forms such as a continuous signal in whichvarious training symbols can be periodically embedded.

The exemplary communication signal 300 can be modulated according to aconstant modulus scheme such as a quadrature amplitude modulated (QAM)scheme. However, it should be appreciated that the modulation scheme ofthe communication signal 300 can vary and can take the form of any knownor later developed modulation scheme, such as BPSK, QPSK, OPSK, FSK, andthe like, without departing from the spirit and scope of the presentinvention.

FIG. 4 is a block diagram of the exemplary OFDM encoder 110 of FIG. 1with an associated data source 410. The OFDM encoder 110 includes a syncpattern generator 420, a training symbol generator 430, a guard symbolgenerator 440 and a number of combining circuits 450.

In operation, the combining circuits 450 can receive a stream of data(payload) symbols from the data source 410 via link 412 and arrange thedata stream into a block of K separate sub-streams of data symbols suchthat each sub-stream of data symbols can be transmitted in a respectiveOFDM sub-band.

Next, the combining circuits 450 can receive a pattern of guard symbolsfrom the guard symbol generator 440 and append the guard symbols to theend of each data sub-stream to produce a block of K sub-streams ofdata/guard symbols.

The combining circuits 450 then can replicate the block of data/guardsymbol sub-streams N number of times, and retrieve N separate patternsof training symbols {T₁, T₂, . . . T_(N)} from the training symbolgenerator 430 via link 432. For each of the N blocks, the combiningcircuits 450 then can replicate each respective pattern of trainingsymbols K number of times and insert the duplicate patterns of trainingsymbols into each of the K separate sub-streams to form N blocks oftraining/data/guard symbol sub-streams with each of the N blocks havinga different pattern of training symbols in each K sub-band.

While the exemplary combining circuits 450 inserts a single pattern oftraining symbols into each of the K sub-streams for a given block, itshould be appreciated that in various embodiments, the training symbolgenerator 430 can provide K×M different patterns of training symbolssuch that the combining circuits 450 can assign each sub-stream oftraining/data/guard symbols a different pattern of training symbols. Instill other embodiments, it should be appreciated that each of the Nblocks can have a number of patterns of training symbols such thatvarious sub-bands can share one of several available patterns oftraining symbols. For example, a given OFDM block having thirty-twosub-streams can receive two patterns of training symbols, replicate eachpattern of training symbols sixteen times and assign each trainingsymbol pattern to alternating sub-streams.

After each block of training/data/guard symbol sub-streams is formed,the combining circuits 450 can then receive a pattern of sync symbolsfrom the sync pattern generator 420, replicate the sync symbols andappend the various training/data/guard symbols to the sync symbols toproduce N blocks of K sub-streams of sync/training/data/guard symbols.

While the exemplary OFDM encoder 110 can arrange various symbols to formfinite blocks of symbols capable of being transmitted as bursts, itshould be appreciated that, in various exemplary embodiments, the OFDMencoder can alternatively form K number of continuous streams of symbolshaving various training and data symbols, without departing from thespirit and scope of the present invention.

FIG. 5 is a block diagram of the exemplary training symbol generator 430of FIG. 4. The training symbol generator 430 has a controller 510, asystem memory 520, a first training sequence generator 530, a secondtraining sequence generator 540 and an input/output interface 590. Thecontroller 510 interfaces with the various other components 520-590using control/data bus 502. While the exemplary training symbolgenerator 430 is depicted as a bussed architecture, it should beappreciated that the functions of the various components 510-590 can beimplemented using various other architectures such as complex circuitsbased on application specific integrated circuits, programmable logicdevices, discrete logic and the like.

In operation, and under control of the controller 510, the input/outputinterface 590 can receive a command to provide N sets of trainingsymbols relating to N separate transmit antennas via link 232 and storethe command in the memory 520. In various exemplary embodiments, theinput/output interface 590 can receive commands or other informationfrom any device such as a wireless transmitter, a wire transmitter, anoptical transmitter, a disc drive, a UART LAN, WAN, parallel digitalinterface, serial digital interface, software interface or any known orlater developed combination of software and hardware without departingfrom the spirit and scope of the present invention.

After the controller 510 imports and stores the command, the controller510 can direct the first training sequence generator 530 to generate afirst set of training symbols t₁[n, k] according to Eq. (15):

t₁[n,k]={S₀,S₁, S₂, . . . S_(K−1)},  (15)

where each S_(i) represents a valid symbol state. As discussed above, aset of training symbols such as t₁[n, k] can be any sequence of symbolsthat is good or otherwise suitable for taking into account factors suchas clock recovery, frequency-shift estimation, peak-to-average signalstrength ratio or any other known or later recognized factor useful forgenerating an advantageous or otherwise adequate training sequence.

The first training sequence generator 530 can generate training symbolsfor communication systems having a constant-modulus signal such that theabsolute value of any training sequence is equal to one (|t₁[n, k]|=1).However, it should be appreciated that, in other exemplary embodiments,the first training sequence generator 530 can generate trainingsequences applicable to any modulation format without departing from thespirit and scope of the present invention.

After the first training sequence generator 530 generates the first setof training symbols t₁[n, k], the controller 510 can provide the firstset of training symbols to the second training sequence generator 540.The second training sequence generator 540 can receive the first set oftraining symbols and generate subsequent sets of training symbolst_(i)[n, k] where i=2, 3, . . . N.

As discussed above, it can be desirable to generate OFDM trainingsequences such that the cross-correlated energy between any trainingsequences transmitted by different antennas is essentially zero, i.e.,Q_(ij)[n]=0. Accordingly, the second training sequence generator 540 cangenerate a second training sequence t₂[n, k] such that thecross-correlation energy between t₁[n, k] and t₂[n, k] is essentiallyzero, i.e., Q₁₂[n]=0. In various exemplary embodiments, such a trainingsequence t₂[n, k] can be generated by replicating and phase-shifting thefirst training sequence t₁[n, k] according to Eq. (16):

$\begin{matrix}{{{t_{2}\left\lbrack {n,k} \right\rbrack} = {{{t_{1}\left\lbrack {n,k} \right\rbrack}{\exp \left( {{- j}\frac{2\pi \; {kl}_{0}}{K}} \right)}} = {{t_{1}\left\lbrack {n,k} \right\rbrack}W_{K}^{- {kl}_{o}}}}},} & (16)\end{matrix}$

for some I_(o) with K_(o)≦K−K_(o). Then it can be directly checked that

q₁₂[n,l]=kδ[l−l_(o)],   (17)

where δ[l] denotes the unit impulse function for a frequency l and l₀ isa reference frequency. Eq. (17) implies that q₁₂[u, l]=0 for|l|≦K_(o)−1, and therefore Q₁₂[n]=0 and Q₂₁[n]=Q₁₂ ^(H)[n]=0.

After the second training sequence generator 540 generates the secondtraining sequence t₂[n, k], it can further generate various othertraining sequences t₃[n, k] t₄[n, k], . . . , t_(N)[n, k] based on thesame principle of Eq.(16) such that Q_(ij)[n]=Q_(ji)[n]=0 for all i≠j.Furthermore, like the first training sequence generator 530, the secondtraining sequence generator 540 can generate t_(i)[n, k]'s such that(|t_(i)[n, k]|=1). In general, for communication systems havingN(N≦K/K₀) transmit antennas, the second training sequence generator 540can generate the remaining N-1 training sequences according to Eq.(18):

t_(i)[n,k]=t₁[n,k]W_(K) ^(−K) ^(o) ^((i-1)k),   (18)

where

${\overset{\_}{K}}_{o} = {\left\lfloor \frac{K}{N} \right\rfloor \geq K_{o}}$

and └x┘ denotes the largest integer no larger than x. Then for twochannels i and j where i<j, the correlation component q_(ij)[n, l] canbe described by Eq.(19):

$\begin{matrix}{{{q_{ij}\left\lbrack {n,l} \right\rbrack} = {\sum\limits_{k = 0}^{K - 1}{{t_{i}\left\lbrack {n,k} \right\rbrack}{t_{j}^{*}\left\lbrack {n,k} \right\rbrack}W_{K}^{- {kl}}}}},} & (19)\end{matrix}$

and by carefully selecting the relative phases between the trainingsequences for different transmit antennas, Eq. (9) above can be reducedto the form of Eq. (20) below:

$\begin{matrix}{\begin{pmatrix}{{\overset{\sim}{h}}_{1}\lbrack n\rbrack} \\{{\overset{\sim}{h}}_{2}\lbrack n\rbrack} \\\vdots \\{{\overset{\sim}{h}}_{i}\lbrack n\rbrack}\end{pmatrix} = {\begin{pmatrix}{Q_{11}\lbrack 0\rbrack} & 0 & \ldots & 0 \\0 & {Q_{22}\lbrack 1\rbrack} & \; & 0 \\\vdots & \; & \ddots & {\vdots \;} \\0 & 0 & \ldots & {Q_{ij}\lbrack n\rbrack}\end{pmatrix}^{- 1}\begin{pmatrix}{p_{1}\lbrack n\rbrack} \\{p_{2}\lbrack n\rbrack} \\\vdots \\{p_{i}\lbrack n\rbrack}\end{pmatrix}}} & (20)\end{matrix}$

Because all off-axis elements of the Q matrix are essentially zero,determining the inverse matrix Q⁻¹ can be greatly simplified.Furthermore, for communication systems using a constant modulusmodulation, i.e., |t_(i)[n, k]|=1, then Q in Eq. (20) can be reduced toQ=K×I, for all i=1, 2, . . . N, and Eq. (20) can be further reduced tothe form of Eq. (21):

$\begin{matrix}{{\begin{pmatrix}{{\overset{\sim}{h}}_{1}\lbrack n\rbrack} \\{{\overset{\sim}{h}}_{2}\lbrack n\rbrack} \\\vdots \\{{\overset{\sim}{h}}_{i}\lbrack n\rbrack}\end{pmatrix} = {\begin{pmatrix}K & 0 & \ldots & 0 \\0 & K & \; & \vdots \\\vdots & \; & \ddots & 0 \\0 & \ldots & 0 & K\end{pmatrix}^{- 1}\begin{pmatrix}{p_{1}\lbrack n\rbrack} \\{p_{2}\lbrack n\rbrack} \\\vdots \\{p_{i}\lbrack n\rbrack}\end{pmatrix}}},{or}} & (21)\end{matrix}$

$\begin{matrix}{\begin{pmatrix}{{\overset{\sim}{h}}_{1}\lbrack n\rbrack} \\{{\overset{\sim}{h}}_{2}\lbrack n\rbrack} \\\vdots \\{{\overset{\sim}{h}}_{i}\lbrack n\rbrack}\end{pmatrix} = {\frac{1}{K}\begin{pmatrix}1 & 0 & \ldots & 0 \\0 & 1 & \; & \vdots \\\vdots & \; & \ddots & 0 \\0 & \ldots & 0 & 1\end{pmatrix}\begin{pmatrix}{p_{1}\lbrack n\rbrack} \\{p_{2}\lbrack n\rbrack} \\\vdots \\{p_{i}\lbrack n\rbrack}\end{pmatrix}}} & (22)\end{matrix}$

Accordingly, the problem of determining the set of channelcharacteristics H_(i)[n, l] for a communication system having N transmitantennas can be reduced to solving Eq. (23):

$\begin{matrix}{{{h_{i}\left\lbrack {n,l} \right\rbrack} = {\frac{1}{K}{p_{i}\left\lbrack {n,l} \right\rbrack}}},} & (23)\end{matrix}$

or alternatively

$\begin{matrix}{{h_{i}\left\lbrack {n,l} \right\rbrack} = {\frac{1}{K}{p_{1}\left\lbrack {n,{l - {\left( {i - 1} \right){\overset{\_}{K}}_{o}}}} \right\rbrack}}} & (24)\end{matrix}$

for I=0, . . . , K _(o)−1 and i=1, 2 . . . , N. Therefore, by carefullyselecting the relative phases between the training sequences fordifferent transmit antennas, the timing sequences for each channel ofeach respective transmit antenna on p_(i)[n, l] are shifted to differentregions in the time domain and the parameters for different channels canbe easily estimated without using a computationally intensive matrixinversion.

As discussed above, for systems where every off-axis element of Q isexactly zero, i.e., Q_(ij)[n]=Q_(ji)[n]=0 for all i≠j, it follows thatQ⁻¹ can be directly replaced by (1/K) I. However, if one or moreoff-axis Q_(ij)[n] are close to, but not exactly, zero, Q⁻¹ may still beapproximated by (1/K) I. That is, it should be appreciated that, invarious embodiments, Q_(ij)[n] can be any value ofQ_(ij)[n]=Q_(ji),[n]≈0 such that Q⁻¹ containing such Q_(ij)[n] can stillbe functionally replaced with (1/K) I, with an understanding that someperformance degradation may occur.

A further advantage to designing the various sets of training symbolsaccording to the above-described technique is that, when Q_(ij)[n]=0 fori≠j as in Eqs. (21) or (22), a channel estimator estimating the variouschannels can effectively attain the theoretical lower mean square error(MSE) boundary. Accordingly, such sets of training symbols can alsoenable an estimator to achieve the best theoretical performance.

While the above-described training symbol design technique can beapplied to various OFDM systems, it should be appreciated that theabove-identified technique can be applied to any other communicationsystem using multiple transmit antennas. Furthermore, the trainingsequences developed above can be easily adapted to pilot sequence designfor pilot symbol aided channel estimation or channel estimation insingle carrier systems. For pilot symbol aided channel estimation inOFDM systems with pilot tones scattered into different times andfrequencies, the pilot sequences can be described as two-dimensional andthe optimum sequence design strategy can accordingly be directly used.Furthermore, the above-described approach can be more flexible thanconventional approaches to pilot symbol aided channel estimation sincethe relative phases of the pilot sequences for different transmitantennas can be shifted in a two-dimensional claim.

FIG. 6 is a block diagram of the exemplary equalizer 160 of FIG. 1. Theexemplary equalizer 160 includes a diversity gain processor 610 and achannel estimator 620 containing a training symbol database 630.

In operation, the diversity gain processor 610 and channel estimator 620can receive various received signals r₁[n, k], r₂[n, k], . . . r_(M)[n,k] such as blocks of symbols from OFDM transmissions via links 152-1,152-2, . . . 152-M. As discussed above, each of the received signalsr_(i)[n, k] can contain multiple transmit signals t_(i)[n, k] accordingto Eq. (7) above transmitted by a plurality of transmit antennas, witheach transmit signal t_(i)[n, k] having an assortment of sync symbols,training symbols, data symbols and guard symbols, as well as any othersymbol types. Also as discussed above, the training symbols embedded inthe communication signals for each of the transmit signals t_(i)[n, k]can be known symbol patterns and formed according to Eqs. (15)-(18)above such that there is essentially no cross-correlated energy betweeneach set of training symbols, i.e. Q_(ij)[n, l]=0 for i≠j.

During operation, channel estimator 620 can extract various sets ofexpected patterns of training symbols t_(i)[n, k] from the trainingsymbol database 630 and process each received signal r_(i)[n, k] basedon the technique described above in Eq. (14) above to derive p_(i)[n,l], which can then be used to determine the various channelcharacteristics based on Eq. (18) or, in the case where a constantmodulus modulation scheme is used, based on Eqs. (21)-(22).

After the channel estimator 620 determines the appropriate set ofchannel characteristics, the channel estimator 620 can export the set ofchannel characteristics to the diversity gain processor 610, which canuse the set of channel characteristics to provide spatial and temporalequalization for the received signals to produce an estimated symbolstream that can be exported to an external device (not shown) via link162.

The exemplary diversity gain processor 610 uses aminimum-mean-square-error (MMSE) technique to provide equalization.However, it should be appreciated that the diversity gain processor 610can use any known or later developed technique useful for temporaland/or spatial equalization without departing from the spirit and scopeof the present invention.

FIG. 7 is a flowchart outlining a first exemplary operation fordeveloping and using optimum training sequences. The process starts instep 710 where a first set of training symbols is developed. Asdescribed above, the first set of training symbols can be any set oftraining symbols that is advantageous or otherwise good forcharacterizing a communication channel. Next, in step 720, adetermination is made as to whether to generate another set of trainingsymbols. If another set of training symbols is to be generated, controljumps to step 770; otherwise, control continues to step 730.

In step 770, a next set of training symbols is generated based on thefirst set of training symbols such that the cross-correlation energybetween the first and second set of training symbols is essentiallyzero. Control then jumps back to step 720 where another determination ismade as to whether to generate another set of training symbols. If moresets of training symbols are to be generated, step 770 is repeated suchthat each new set of training symbols generated will not appreciablyhave any cross-correlation energy with any of the previously generatedsets of training symbols. The exemplary technique generates subsequenttiming sequences by phase shifting the first time sequence according toEqs. (13)-(16) above. However, it should be appreciated that, in variousexemplary embodiments, the particular technique used to generatesubsequent sets of training symbols can vary without departing from thespirit and scope of the present invention.

In step 730, because no further sets of training symbols are to begenerated, the training sequences generated in step 710, 720 and 770 aretransmitted. The exemplary sets of training symbols are transmittedaccording to an OFDM paradigm with the different sets of trainingsymbols transmitted using different transmit antennas. However, asdiscussed above, it should be appreciated that the various sets oftraining symbols can be transmitted according to any other known orlater developed paradigm using multiple transmit antennas and/orsub-bands without departing from the spirit and scope of the presentinvention.

Next, in step 740, the training symbols are received. While theexemplary technique uses multiple OFDM receivers coupled to anequalizer, it should be appreciated that the number of OFDM receiverscan vary without departing from the spirit and scope of the presentinvention. Furthermore, as discussed above, while the exemplary receivercan operate according to an OFDM paradigm, it should be appreciated thatthe receiver can operate according to any other known or later developedparadigm without departing from the spirit and scope of the presentinvention.

Then, in step 750 the communication channels between the varioustransmit antennas and receive antennas can be estimated. As discussedabove, the exemplary technique can estimate the various channels using acorrelation matrix Q having all of the off-diagonal elements equal tozero to make the process of channel estimation extremely simple.Furthermore, as discussed above, in systems using a constant modulusmodulation approach, the correlation matrix Q can be reduced to theidentity matrix, thereby alleviating the necessity for performing aninverse transform altogether. Control then continues to step 760 wherethe operation stops.

As shown in FIG. 1-6, the systems and methods of this invention arepreferably implemented on a digital signal processor (DSP) or otherintegrated circuits. However, the systems and methods can also beimplemented using any combination of one or more general purposecomputers, special purpose computers, program microprocessors ormicrocontroller and peripheral integrating circuit elements, hardwareelectronic or logic circuits such as application specific integratedcircuits (ASICs), discrete element circuits, programmable logic devicessuch as PODs, POAs, FPGAs, PALs, or the like. In general, any device onwhich exists a finite state machine capable of implementing the variouselements of FIGS. 1-6 and the flowchart of FIG. 7 can be used toimplement the training sequence functions.

While this invention has been described in conjunction with the specificembodiments thereof, it is evident that many alternatives, modificationsand variations will be apparent to those skilled in the art.Accordingly, preferred embodiments of the invention as set forth hereinare intended to be illustrative, not limiting. There are changes thatmay be made without departing from the spirit and scope of the presentinvention.

What is claimed is:
 1. A method for communicating, comprising:transmitting, via a first transmitter, a set of first known symbolsusing a first communication channel; and transmitting, via a secondtransmitter, a plurality of sets of second known symbols using a secondcommunication channel; wherein the plurality of sets of second knownsymbols is based on the set of first known symbols and across-correlation estimate between the first set of known symbols andone of the plurality of sets of second known symbols is zero, where achannel estimation is performed based on the set of the first knownsymbols, without performing a matrix inversion.